Compositional Covariate Importance Testing via Partial Conjunction of Bivariate Hypotheses

Lucas Janson; Ritwik Bhaduri; Siyuan Ma

REVIEW 3 major objections 4 minor 90 references

Compositional Covariate Importance Testing via Partial Conjunction of Bivariate Hypotheses

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Compositional covariates defeat standard importance tests; this paper defines importance as the unique nontrivial Markov boundary and tests it via partial conjunction of bivariate conditional hypotheses.

desk verdict Novel and valuable theory for compositional covariate importance, but the power-advantaged methods rely on an unproven PRDS assumption and two appendix proofs need fixing. read the letter →

arxiv 2501.00566 v1 pith:BPGHAICO submitted 2024-12-31 stat.ME math.STstat.TH

Ritwik Bhaduri , Siyuan Ma , Lucas Janson This is my paper · ORCID

classification stat.MEmath.STstat.TH MSC 62H1562G10

keywords compositionaldataMarkovboundarybivariateconditionalindependencepartialconjunctionhypothesismultipletestingfalsediscoveryratefamilywiseerrorrandomizationtest

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Compositional covariates arise whenever a total is split into parts, such as microbiome relative abundances, time-use budgets, or voting shares, and they silently break standard regression tools: each covariate is a deterministic function of the others, so every conditional importance test is trivially true and every marginal test is trivially false. This paper establishes that a workable notion of importance still exists: under mild conditions, the unique nontrivial Markov boundary of the response exists and equals the set $S$ of covariates $j$ for which every bivariate conditional hypothesis $Y \perp\!\!\perp X_{i,j} \mid X_{i,j}^c$ (for all $i \neq j$) is false. It then shows that testing membership in $S$ reduces to a partial conjunction hypothesis test, and builds single-test and multiple-testing procedures called BCP with proven FWER and FDR control. Simulations indicate the methods are valid and powerful across Dirichlet, logistic-normal, sparse-compositional, and even non-compositional covariate distributions.

What carries the argument

The load-bearing object is the bivariate conditional independence hypothesis $H_{i,j}: Y \perp\!\!\perp X_{i,j} \mid X_{i,j}^c$, which is non-degenerate under compositionality because conditioning on all but two coordinates leaves a random pair. The set $S$ aggregates these hypotheses, and Theorem 2.1 connects $S$ to the Markov boundary via an iterated set operation $(\Delta \circ)^k I$ built from an intersection-property lemma for supports with a single equivalence class. On the testing side, the paper's methods are partial conjunction hypothesis (PCH) tests — hypotheses stating that fewer than $r$ of a collection of base hypotheses are false — applied to the bivariate p-values, combined with Bonferroni or Simes global tests, and then wrapped in Holm-style or Benjamini-Hochberg multiple testing procedures to form the BCP family. This machinery converts an untestable definition (a Markov boundary on a measure-zero support) into a testable composite of ordinary conditional independence tests.

What would settle it

Generate compositional covariates with a known sparse Markov boundary (e.g., $Y = X_1 + \varepsilon$, $X$ Dirichlet) but force the bivariate p-values to be negatively dependent — for instance by using antithetic resampling in the conditional randomization test — and apply BCP($\bar{s}$)-BH at FDR 10%. If the empirical FDR exceeds 10%, Theorem 3.3's Simes-based guarantee is refuted. Alternatively, a distribution like Example 2 where the path-connectivity condition of Corollary 2.1 fails should produce multiple nontrivial Markov boundaries, checking the uniqueness claim.

Watch

Extended reading notes

Core claim

The paper's central theoretical result is Theorem 2.1: for compositional $X$, if $S=[p]$ then no nontrivial Markov boundary exists, and otherwise, provided $S^c \in (\Delta \circ)^k I$ for some $k$, $S$ is the unique nontrivial Markov boundary, where $H_{i,j}: Y \perp\!\!\perp X_{i,j} \mid X_{i,j}^c$, $I=\{\{i,j\}: H_{i,j} \text{ true}\}$, and $\Delta$ collects pairs of sets for which an intersection-property lemma applies. Under Assumption 1, each hypothesis $H_{0j}: j\notin S$ equals a partial conjunction hypothesis: fewer than $r$ of the bivariate nulls for that $j$ are false, and any strict upper bound $\bar{s}>|S|$ yields a valid test. The paper proves validity of Bonferroni-based BCP tests under arbitrary dependence, and Simes-based BCP tests under a PRDS condition on the base p-values; it also proves FWER control for Holm-style BCP algorithms and FDR control for a Benjamini-Hochberg BCP algorithm.

Load-bearing premise

The Simes-based BCP methods, which carry the main power advantages and are used in the primary simulations, are valid only if the bivariate base p-values are positively regression dependent on each null subset (PRDS); the paper calls this plausible but not easily provable, and supplies no concrete conditional-independence test with a proven PRDS guarantee.

Editorial extensions

If this is right

Standard conditional-independence-based variable selection and testing, including parametric coefficient tests, knockoffs, and conditional randomization tests, has provably trivial power on compositional covariates; BCP methods restore nontrivial, error-controlled inference.
When an upper bound $\bar{s}$ on the number of important covariates is known (for example $\bar{s}=p/2$), BCP tests are substantially more powerful than the always-valid default $\bar{s}=p-1$.
The scope extends beyond compositional vectors: the same theory and procedures apply to covariates satisfying any single deterministic constraint, including linear subspaces or the unit sphere.
Conditioning on covariates that are a priori sparse (Theorem 3.4) recovers power without sacrificing validity, as long as $|S^c \cap D| \neq 1$.
In non-compositional regression settings, BCP methods retain most of the power of state-of-the-art univariate conditional independence tests, so the same toolbox transfers without loss.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The PRDS assumption could be verified for concrete conditional-independence tests: for instance, the distilled conditional randomization test with Gaussian designs may satisfy PRDS under exchangeable resampling, which would upgrade Corollary 3.2 and Theorem 3.3 from plausible to proven.
The same PCH-of-bivariate-hypotheses construction might extend to multiple deterministic constraints (e.g., $k$ constraints) by testing $k$-variate conditional independence, but at $O(p^{k+1})$ hypotheses; the paper's screening speedups suggest a possible path to tractability.
For experimental design, Remark 2 shows the Markov boundary can depend on the support of $X$, not just on $Y\mid X$; this implies that the choice of design partially determines the scientifically meaningful target of selection.
The paper's methods could be used as a diagnostic for whether a compositional regression problem has any parsimonious structure: if $S=[p]$ (all bivariate tests false), no nontrivial Markov boundary exists and all covariates are effectively important.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Rough take: the paper earns a serious referee. The central characterization—that under compositionality the unique nontrivial Markov boundary is exactly the set S defined by bivariate conditional independence hypotheses—is new and non-obvious, and the Bonferroni-based BCP procedures built on it are rigorously valid under arbitrary dependence. That alone is a real contribution to a problem where standard conditional testing is degenerate and unconditional testing is unreliable.

The paper is also honest about its main weakness. The Simes-based BCP(s), BCP(s)-BH, and BCP(s)-Holm—the methods used in all main-text simulations and recommended for power—require the base p-values to be PRDS. The authors explicitly say this is "plausible (if not easily provable)" and offer no conditional independence test that provably satisfies it. The dCRT p-values share response and overlapping covariates, so their dependence is complex; the Section 3.5 screening speedups further alter it. If PRDS fails, the Simes-based p-values and the BH/Holm procedures can be liberal. The Bonferroni versions remain valid, so the core idea survives, but the advertised power gains are conditional on an unverified distributional conjecture.

Two appendix proofs also have gaps. In Corollary 2.1's proof, the constructed intermediate point w† needs to lie in the simplex; the inequality shown only gives ≤1, not ≤ 1 − sum of conditioned coordinates, so the point can have a negative coordinate. Corollary 2.2's Step 4 appears to condition on a larger set than the one implied by the preceding weak-union and intersection steps—likely a typo, but as written the proof doesn't go through. These are fixable but mean the simplified conditions for continuous and factor covariates are not fully established as stated.

The simulations are extensive, the code is shipped, and the comparison with LOO and AdaFilter is informative. I'd send this to peer review with a request for major revision: either prove PRDS for a concrete test or relegate the Simes methods to a clearly labeled conjecture, fix the two proof details, and re-run the power claims if needed. The paper is for statisticians working on compositional or constrained covariate inference; they will want to read it.

Referee Report

3 major / 4 minor

Summary. The paper addresses covariate importance testing when covariates are compositional. Because each compositional covariate is a deterministic function of the others, standard conditional independence hypotheses are degenerate, and unconditional hypotheses are all false in typical sparse models. The authors define the set of important covariates via the Markov boundary, prove under conditions that it coincides with the set S of indices j for which every bivariate conditional independence hypothesis H_{i,j}: Y ⊥⊥ X_{i,j} | X_{i,j}^c is false, and then test membership in S through partial conjunction hypotheses built from bivariate p-values. They propose BCP procedures: Bonferroni-based tests valid under arbitrary dependence, and Simes-based tests and BH/Holm multiple testing procedures whose validity requires a PRDS assumption on the base p-values. Numerical experiments compare these methods with leave-one-out and univariate benchmarks in compositional, non-compositional, and sparse compositional settings.

Significance. If the results hold, this is a valuable contribution: it gives a principled, falsifiable definition of relevant compositional covariates and connects it to a practical testing framework. The Markov-boundary characterization in Theorem 2.1 and the Bonferroni-based error control results are rigorous under the stated assumptions, and the paper includes reproducible code and extensive simulations. The Janus-faced nature of conditional and unconditional hypotheses under compositionality is clearly explained. However, the power-oriented methods advertised and used throughout the main text rest on an unverified PRDS assumption, and two appendix proofs that support the main simplifying corollaries contain gaps. The significance is therefore high conditional on repair of these issues.

major comments (3)

[Section 3.1, Corollary 3.2, Theorems 3.2 and 3.3] The Simes-based BCP procedures, which are the methods used in all main-text simulations and recommended for their power, are valid only if the base p-values are positively regression dependent on each null subset (PRDS). The manuscript itself says this is "plausible (if not easily provable)" and provides no conditional independence test, including the dCRT used in the simulations, for which PRDS is actually established. The dCRT p-values share the same response and overlapping conditioning sets, so their joint dependence is complex and not obviously PRDS. Moreover, Section 3.5's data-dependent screening changes the dependence structure, and the paper concedes it "could in principle impact the validity" of the positive-dependence-based procedures. Since the advertised power advantages and the FWER/FDR guarantees for BCP(s)-Holm and BCP(s)-BH rest on this unverified assumption, I request either a proven PRDS result for a concrete test class (e.g., the dCRT under the Gaussian or Dirichlet simulation models) or a restructuring that presents the Bonferroni-based procedures as the formally guaranteed methods and the Simes-based procedures as empirically supported heuristics.
[Appendix B.3, proof of Corollary 2.1] The constructed point w† in Step 3 can fail to lie in the simplex, so the claimed equivalence proof is incomplete. In the first case, the coordinate of w† corresponding to A∩B is 1 - sum_{j in A\B} w_j - w*_{B\A} - sum c_j, and the displayed inequality does not prevent this coordinate from being negative. The subsequent claim f_X(w†) > 0 follows only if w† is in the relevant ball within the simplex slice, which is not established. This gap affects Corollary 2.1, the main simplification of Theorem 2.1 for continuous compositional distributions, and needs to be repaired before the corollary can be considered proven.
[Appendix B.4, proof of Corollary 2.2] The proof of Corollary 2.2 for factor covariates has a serious gap. In Step 4, after applying Lemma 2.2 to A = S_F^c ∩ F_k and B = M^c ∩ F_k^c, the displayed conditioning set and the claimed blanket are not coherent: the set (S_F ∩ F_k) ∪ (M ∪ F_k^c) is not generally a subset of M, so it cannot contradict the minimality of the Markov boundary M. The derivation of Y ⊥⊥ X_{S_F^c ∩ F_k} | X_{(S_F ∩ F_k) ∪ F_k^c} also needs justification. As stated, Corollary 2.2 is not proven and should either be supplied with a corrected argument or explicitly deferred.

minor comments (4)

[Figure 1 caption] The word "selelction" should be "selection".
[Corollaries 2.1 and 2.2, assumption (i)] The condition "for all i, j in S^c" should require i ≠ j, since H_{i,i} is not defined. The same comment applies to Corollary 2.2 and Corollary C.1.
[Appendix B.3, Step 2] The text "0 ≥ c < 1" appears to be a typo and should read "0 ≤ c < 1".
[Section 3.1, Eq. (2) and (3)] The notation P_{(i),j} is used for order statistics of the p-values in column j, but the indexing in the displayed equations is not defined explicitly; a sentence defining P_{(i),j} as the ith smallest of {P_{i',j} : i' ≠ j} would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the S/H_{i,j} connection is definitional but externally anchored by the Markov boundary; the main validity gap (unproven PRDS for Simes-based BCP) is an assumption, not a circular reduction.

full rationale

The derivation chain is self-contained. S is defined as {j: H_{i,j} false for all i≠j}, and H_{0j}: j∉S is exactly H^{p-1}_{0j}, a partial conjunction of the same bivariate CI hypotheses; this is a definitional identity used to build a test, not a prediction smuggled in. The external anchor is Theorem 2.1, proved in Appendix B.2 from Lemma 2.1 and Proposition B.1, showing S coincides with the unique nontrivial Markov boundary under the (Δ∘)^k I condition; the proof adapts Peters (2015), not self-citation. Corollaries 2.1/2.2/C.1 supply explicit sufficient conditions. The BCP validity results (Corollary 3.1, Theorems 3.1–3.3) follow from Benjamini–Heller/Bogomolov PCH theory applied to user-supplied valid base p-values; no parameter is fitted to data to manufacture a prediction. Self-citations (Ma et al. 2024, Liu et al. 2022) are used only for comparison or as an external test choice. The one caveat worth recording is that the Simes/BH procedures advertised for power require PRDS of the base p-values, which the paper explicitly labels 'plausible (if not easily provable)' (Section 3.1); if PRDS fails, those procedures may not control error rates. That is a substantive statistical assumption/limitation, but it is not a circular reduction of the paper's conclusions to its inputs. Data-dependent screening (Section 3.5) is similarly a caveat about dependence, not circularity. Overall: very low or no circularity burden; score 1 reflects only minor self-citation and definitional proximity, neither load-bearing.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim rests on domain assumptions about compositional support, uniqueness of the Markov boundary (Assumption 1), validity of the base CI tests, and, for the recommended Simes procedures, PRDS of base p-values. User-chosen inputs such as the upper bound s and resample count are not fitted parameters. No new physical entities are introduced.

free parameters (3)

s (strict upper bound on |S|) = p-1, p/2, or s+1 in simulations
User-specified input to BCP methods; smaller s gives more power but requires domain knowledge. Not fitted to data in the theory.
K (number of dCRT resamples) = 1500 (or 25,000 in one appendix)
Monte Carlo resample count for base p-values; affects resolution of p-values, not a model parameter.
Screening p-value threshold for speedups = 0.1
P-values above this are set to 1 to save computation; chosen by hand in Appendix E.9.

assumptions (7)

domain assumption X is compositional: sum_j X_j = 1 and X has no additional deterministic constraints beyond this (except the disjoint per-factor constraints in Section 2.3.2).
Assumed throughout; Remark 1 notes that extra constraints would require trivariate or higher-order hypotheses.
domain assumption Assumption 1: either S = [p] and no nontrivial Markov boundary exists, or S is the unique nontrivial Markov boundary.
Adopted in Section 3 before constructing tests; needed for the identity H_0j = H^{s+1}_{0j}.
domain assumption The bivariate conditional independence tests producing P_{i,j} are valid (super-uniform under H_{i,j}).
Section 3.1 and 3.4 assume a valid CI test is available; dCRT in the simulations requires knowing or sampling the conditional distribution of X_{i,j}|X_{i,j}^c.
domain assumption For the Simes-based methods, the null base p-values are PRDS.
Stated in Corollary 3.2, Theorem 3.2, Theorem 3.3; not proven for any concrete CI test, only empirically supported.
domain assumption The technical conditions of Corollary 2.1 (path-connectivity and graphical connection of true nulls) hold for continuous compositional distributions used in practice.
Used to instantiate Theorem 2.1 for Dirichlet and logistic-normal covariates.
standard math The weak union and intersection properties of conditional independence, including the Peters (2015) version for non-Lebesgue supports, hold.
Used in Lemma 2.1, Lemma 2.2, and several appendix proofs.
standard math Partial conjunction testing theory (Benjamini and Heller 2008) and the FDR result of Bogomolov (2021) apply as stated.
The BCP p-values and FDR control argument are inherited from these results.

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Cite this review

Pith. "Pith review of Compositional Covariate Importance Testing via Partial Conjunction of Bivariate Hypotheses." pith.science (2026). https://pith.science/paper/BPGHAICO

@misc{pith2026250100566,
  author       = {Pith},
  title        = {Pith review of: Compositional Covariate Importance Testing via Partial Conjunction of Bivariate Hypotheses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPGHAICO}},
  note         = {Machine review of arXiv:2501.00566}
}

read the original abstract

Compositional data (i.e., data comprising random variables that sum up to a constant) arises in many applications including microbiome studies, chemical ecology, political science, and experimental designs. Yet when compositional data serve as covariates in a regression, the sum constraint renders every covariate automatically conditionally independent of the response given the other covariates, since each covariate is a deterministic function of the others. Since essentially all covariate importance tests and variable selection methods, including parametric ones, are at their core testing conditional independence, they are all completely powerless on regression problems with compositional covariates. In fact, compositionality causes ambiguity in the very notion of relevant covariates. To address this problem, we identify a natural way to translate the typical notion of relevant covariates to the setting with compositional covariates and establish that it is intuitive, well-defined, and unique. We then develop corresponding hypothesis tests and controlled variable selection procedures via a novel connection with \emph{bivariate} conditional independence testing and partial conjunction hypothesis testing. Finally, we provide theoretical guarantees of the validity of our methods, and through numerical experiments demonstrate that our methods are not only valid but also powerful across a range of data-generating scenarios.

Figures

Figures reproduced from arXiv: 2501.00566 by the authors.

**Figure 2.** Comparison of methods for (a) single testing and (b) multiple testing with [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Effect of conditioning out the sparse covariates on power for single testing (left) [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Comparison of FWER and average power for variable selection with Dirichlet [PITH_FULL_IMAGE:figures/full_fig_p054_4.png]

**Figure 5.** Figure 5: Comparison of methods for (a) single testing, (b) FWER control and (c) FDR [PITH_FULL_IMAGE:figures/full_fig_p055_5.png]

**Figure 6.** Figure 6: Comparison of FWER and average power for variable selection with multivari [PITH_FULL_IMAGE:figures/full_fig_p056_6.png]

**Figure 7.** Figure 7: Effect of conditioning out the sparse covariates on type I error (left), FWER [PITH_FULL_IMAGE:figures/full_fig_p057_7.png]

**Figure 8.** Figure 8: Simulation studies showing the effects of sampling from an estimated Dirichlet [PITH_FULL_IMAGE:figures/full_fig_p058_8.png]

**Figure 9.** Figure 9: Comparison of BCP(s) with Simes and Bonferroni PC p-values. 61 [PITH_FULL_IMAGE:figures/full_fig_p061_9.png]

**Figure 10.** Figure 10: Comparison of BCP(s) with Simes and Bonferroni PC p-values for Logisticnormal covariates. 62 [PITH_FULL_IMAGE:figures/full_fig_p062_10.png]

**Figure 11.** Figure 11: Comparison of BCP(s) with Simes and Bonferroni PC p-values for multivariate normal covariates. 63 [PITH_FULL_IMAGE:figures/full_fig_p063_11.png]

**Figure 12.** Figure 12: Comparison of effect of conditioning out sparse covariates on BCP( [PITH_FULL_IMAGE:figures/full_fig_p064_12.png]

**Figure 13.** Figure 13: Performance of AdaFilter on Dirichlet data in controlling (a) FWER and (b) [PITH_FULL_IMAGE:figures/full_fig_p065_13.png]

**Figure 14.** Figure 14: Comparison of proposed method with computational speedups discussed in [PITH_FULL_IMAGE:figures/full_fig_p066_14.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed August 10, 2026 · model on record in the stance chip above.

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